A simple model for the location of Saturn's F ring Luis Beneta,∗, Angel` Jorbab aInstituto de Ciencias F´ısicas, Universidad Nacional Aut´onomade M´exico (UNAM) Apdo. Postal 48{3, 62251 Cuernavaca, Mexico bDepartament de Matem`atica Aplicada i An`alisi,Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain Abstract In this paper, we introduce a simplified model to understand the location of Saturn's F ring. The model is a planar restricted five-body problem defined by the gravitational field of Saturn, including its second zonal harmonic J2, the shepherd moons Prometheus and Pandora, and Titan. We compute accu- rate long-time numerical integrations of (about 2.5 million) non-interacting test-particles initially located in the region between the orbits of Prometheus and Pandora, and address whether they escape or remain trapped in this re- gion. We obtain a wide region of initial conditions of the test particles that remain confined. We consider a dynamical stability indicator for the test particles' motion defined by computing the ratio of the standard deviation over the average value of relevant dynamical quantities, in particular, for the mean-motion and the semi-major axis. This indicator separates clearly a subset of trapped initial conditions that appear as very localized stripes in the initial semi-major axis and eccentricity space for the most stable or- bits. Retaining only these test particles, we obtain a narrow eccentric ring which displays sharp edges and collective alignment. The semi-major axis of the accumulation stripes of the stable ring-particles can be associated with resonances, mostly involving Prometheus' outer Lindblad and co-rotation resonances, but not exclusively. Pandora's inner Lindblad and co-rotation resonances as well as low-order three-body resonances typically coincide with arXiv:1612.01451v1 [astro-ph.EP] 5 Dec 2016 gaps, i.e., regions of instabilities. Comparison of our results with the nominal data for the F ring shows some correspondence. ∗Corresponding author Email addresses: [email protected] (Luis Benet), [email protected] (Angel` Jorba) Preprint submitted to Elsevier July 6, 2018 Keywords: Saturn, rings, Planetary rings, Resonances, rings 1. Introduction Saturn's F ring is a fascinating narrow ring with a rich time-varying structure, which has puzzled dynamical astronomers since its discovery by the Pioneer 11 team in 1979 (Gehrels et al., 1980). It is located outside Saturn's A ring, close but beyond Roche's limit for ice, and is believed to be the result of the ongoing action of competing accretion and disruptive pro- cesses (Barbara and Esposito, 2002). The F ring is narrow, non-circular and inclined, has azimuthal dependent properties which may change on time, and displays certain localized radial structures (Porco et al., 2005). It consists of a dense core (1−40 km) embedded in a broader belt of dust (∼ 700 km), with additional separated dusty components named strands, the most prominent form one-arm kinematic spirals on either side of the core (Charnoz et al., 2005). It also contains an underlying belt of forming moonlets (Cuzzi and Burns, 1988) that produce the observed \fan" structures (Murray et al., 2008; Beurle et al., 2010), and whose collisions with the core manifest as jets (Mur- ray et al., 2008). For recent reviews see Colwell et al. (2009, sect. 13.5) and Charnoz et al. (2009). The F ring is perturbed by the shepherd moons Prometheus and Pandora, that orbit on either side of the ring. The discovery of these moons (Smith et al., 1981, 1982) was initially interpreted as the success of the confine- ment produced by the shepherding mechanism, proposed by Goldreich and Tremaine (1979). Yet, the torques from the shepherd moons do not balance out at the location of the ring (Showalter and Burns, 1982). Moreover, the shepherd moons orbit around Saturn on seemingly chaotic orbits (Poulet and Sicardy, 2001; French et al., 2003; Goldreich and Rappaport, 2003a,b). While it is clear that Prometheus and Pandora play an important role on the dynamics and structure of the ring (Showalter and Burns, 1982; Winter et al., 2007), it is not clear what is the actual mechanism that keeps the ring confined at its location (Esposito, 2006). Some of the structural phenomena described above have been analyzed previously through numerical simulations. For instance, using independent test particle models with periodic boundary conditions along the azimuthal direction, Giuliatti Winter et al. (2000) integrated the equations of motions 2 of the circular restricted three-body problem of Saturn and Prometheus up to a few tens of revolutions. They showed that, after a close approach with this moon, ring particles initially located at the strands of the F ring were scattered inwards and outwards, forming gaps and waves; these were later confirmed by Cassini, and were named channels (Porco et al., 2005). These calculations were taken further (Murray et al., 2005), concluding that stream- ers and channels are part of the same phenomenon and can be understood in terms of the gravitational interaction with this moon and its eccentric motion. Other calculations considered the effect of Pandora during 160 yr, and concluded that the motion of embedded moonlets in the the ring is likely chaotic, removing it from the F ring region (Giuliatti Winter et al., 2006). Charnoz et al. (2005) discovered a kinematic spiral strand and interpreted it, based on numerical simulations spanning 2000 orbital periods, as the effect of interactions with small satellites in the F ring region. More complex inte- grations, including 14 massive Saturn moons and spanning tens of thousands of Prometheus periods, were conducted by Cuzzi et al. (2014), and led them to the conclusion that certain regions of stability arise because the pertur- bations induced by an encounter with Prometheus are counter-balanced by subsequent encounters with the same shepherd. While these investigations have clarified the influence of the shepherd moons in the variations on the structure of the F ring, the question of its confinement remains unanswered. The question can be restated in terms of the location of the ring. The fact that the shepherd moons move on chaotic trajectories makes this problem more interesting since, strictly speak- ing, it breaks the periodicity of the perturbations. With regards to this, it is worth quoting Tiscareno (2013), who points out that, despite the time- varying clumpy and kinky structure, the F ring core \maintains over decadal timescales the shape of a freely precessing eccentric inclined ellipse; the orbital solution formulated to account for Voyager and other pre-Cassini data (Bosh et al., 2002) has, somewhat surprisingly, remained a good predic- tor of the core's position through the Cassini mission". In this paper we address the question of the confinement and location of Saturn's F ring within an independent (non-interacting) test particle model. We consider a simplified planar restricted five-body model and follow the dy- namics of a large number of test particles (∼ 2:5 millions) up to 6×106 orbital revolutions of Prometheus, slightly more than 10000 years. The model in- cludes the gravitational field of Saturn with its second gravitational harmonic J2, Prometheus, Pandora and Titan; preliminary results appear in (Benet 3 and Jorba, 2013). We find a broad set of test-particle initial conditions that remain trapped within the region between Prometheus and Pandora. Within this set, there are initial conditions that remain well-localized and correspond to the more stable ones with respect to their radial excursions. A projection onto the X − Y space of a snapshot of this stable subset yields a narrow and eccentric ring with collective alignment, whose location properties and width can be compared with the observations. The model we consider is simple in the sense that it does not include contributions which are important for a detailed realistic description, such as the J4 gravitational harmonic or the gravitational interaction of other Sat- urn moons. The assumption of non-interacting test particles is equivalent to neglecting any ring particle collisions and self-gravity. While the assump- tions are rather strong, in particular for the time-span of our integrations, our view is that the physics of the existence of the F ring, or other phenom- ena, is not related to matching very specific parameters, but because certain necessary conditions are fulfilled. In the present case, the important property is the existence of phase-space regions where the radial diffusion is strongly suppressed. The paper is organized as follows: In section 2 we describe of our general approach and introduce the simple model that we study. Section 3 describes the numerical results obtained, where we focus first on the test particles that remain trapped up to a maximum time, and then classify them dy- namically according to their stability. The stability analysis is based on a dimensionless dynamical indicator related to the radial excursions. Filtering the orbits that do not satisfy a stability condition yields a narrow, eccen- tric, sharp-edged ring, whose semi-major axis and width are then compared with the observations. We relate the accumulation in the semi-major axis of the ring particles, and the gaps between them, to orbital resonances, which involve mainly Prometheus outer Lindblad and co-rotation resonances, but not exclusively. Finally, in section 4 we summarize our approach and results. 2. A simplified model for Saturn's F ring: the scattering approach A complete description of the dynamics of Saturn's F ring must con- tain the gravitational interactions of Saturn including its flattening, the in- fluence of all Saturn's moons including in particular the shepherd moons, Prometheus and Pandora, and the interactions among the ring particles themselves.